The present invention relates to an analysis method and an analyzer for implementing the method, intended to analyze and simulate various periodic phenomena such as, mechanical vibrations, or variations in temperature, pressure, fluid flows, electric currents or electric voltages, or similar.
Though Laplace transformation has been traditionally used in frequency response analysis in particular, up until now step response analysis has had to resort to inaccurate graphical methods.
The testing of step response includes the recording of the dynamic behavior of an extremely simple test technique, as it is easy to use simple test equipment to suitably develop the required signals.
The level of difficulty of interpreting the experimental step responses depends of the level of the tested oscillator. A complex step response includes an additional series of time-dependent terms in the Laplace transfer function. As the number of terms is the level of the system, it is easy to show that, when the level of the system rises, it becomes increasingly difficult to use graphical means to resolve the response into components. If the components cannot be defined, it will be impossible to carry out quantitative calculations.
In typical first-degree growth and damping responses, the time constant is usually measured by taking the time required for the response to increase to 63% of its final size and damping to 37% of the initial value.
The interpretation of second or higher degree responses depends on the degree of damping present. In cases of weak or mediocre damping in responses of the second order, the damped specific angular velocity can be calculated, according to the state of the art, by measuring the oscillation period, as shown in FIGS. 1 and 2. Thus, xcfx89t=2xcfx80/p. Frequency ft=xcfx89/2xcfx80.
FIGS. 3 and 4 show an alternative technique according to the state of the art for calculating the damping factor in weak and mediocre cases. FIG. 3 shows graphically the natural logarithm of each maximum and minimum, on the basis of the numerical position of the series of maximums and minimums, while FIG. 4 shows a standard curve of the peak maximums in relation to the damping factor, which is most suitably used in the central areas. The non-damped specific angular velocity can be obtained from equation (1):                               ω          n                =                                            ω              l                                                      1                +                                  ζ                  2                                                              .                                    (        1        )            
FIGS. 5 and 6 shows a case of strong damping, for which the most precise technique is to construct a tangent to the point of inflexion. The position of this tangent can be applied to the corresponding tangent for each standard square response curve. Interpolation can be used where necessary to obtain the most suitable non-damped angular velocity xcfx89n and damping xcex6 values from the curve.
The graphical interpretation of higher-degree step responses can be highly unreliable or even impossible. While it may be possible to identify the transfer function by appropriately using a computer and applying numerical analysis and curve fitting, it is usual to use the more comprehensive frequency response analysis, instead of analyzing the step response.
The present invention is intended to eliminate the many defects referred to above in the analysis of step or other response, and to allow rapid and precise response analysis, so that the analysis will be able to be used in many fields of technology, in which the deficient art has so far prevented its use.
Thus, the method of analysis in this invention has been developed to solve the problems in the characteristic manner disclosed in the accompanying Claims.
The general feature of the invention can be described briefly as follows.
According to the invention, the measured step response is transformed into an illustrative form, by means of a new integral transformation. Whereas up until now Laplace transformation in particular has been used when analyzing frequency response, and imprecise graphical methods have been resorted to in cases of step response, the transformation used in this invention can be called Fourier-Laplace transformation.
Step response, which can sometimes be extremely complex, can be expressed explicitly using the transformation referred to above, with the aid of a mathematical function. Whereas so far it has only been possible to analyze the step responses created by a single damping sine wave, it is possible, according to the invention to form a series of simple damping sine waves and use this as a base for analysis. Each term in the series is the solution of a linear differential equation, so that an unknown function can be expressed using superposition, i.e. as the sum of the simple sine waves that have been discovered. The analysis expresses each term""s angular velocity xcfx89, damping xcex6, starting time ts and steady-state value Y∞, thus completely defining the damping sine wave.
This method can be used to analyze practically any step response whatever, simulate it precisely, and state the final result as a mathematical formula. Thus, the invention need not be limited to any specific technical application, but can be applied in very many fields of technology, medical science, and to other applications. The invention also allows a future transfer to the use of step response analysis, which is considerably cheaper than frequency response to use.
The accompanying drawings illustrate the invention and the state of the art. The state of the art has already been described above.